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Instantons and jumping lines. (English) Zbl 0596.32031
The author investigates the moduli space of SV(2)-instantons of topological charge n. By means of the Ward transform this is equivalent to the space of certain holomorphic bundles on \({\mathbb{C}}{\mathbb{P}}_ 3\) and these bundles may be described by monads. Recently, S. Donaldson [Commun. Math. Phys. 93, 453-460 (1984; Zbl 0581.14008)] has identified this space of monads with the moduli space of semi-stable rank 2 holomorphic vector bundles on \({\mathbb{P}}_ 2\). Blowing up one point in \({\mathbb{P}}_ 2\) gives a ruled surface on which the pull-back of a semi- stable bundle will jump at n lines of the ruling. The author carefully analyzes this jumping behaviour so that one may reconstruct the given bundle from suitable jumping data at these distinguished lines. This gives him enough information to be able to calculate the fundamental group of the original moduli space: \({\mathbb{Z}}_ 2\) for n even and 0 for n odd. The moduli space of stable bundles is much easier to work with than the monad description especially because the reality conditions have disappeared. This paper is a good illustration of this fact. This was also Donaldson’s motivation of course (which immediately allowed him to conclude connectivity of the moduli spaces).
Reviewer: M.Eastwood

32G05 Deformations of complex structures
32L05 Holomorphic bundles and generalizations
53C80 Applications of global differential geometry to the sciences
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32L25 Twistor theory, double fibrations (complex-analytic aspects)
57S30 Discontinuous groups of transformations
Full Text: DOI
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